# Recognizing connectedness from vertex-deleted subgraphs

Bowler, Andrew and Brown, Paul and Fenner, Trevor and Myrvold, W.
(2010)
Recognizing connectedness from vertex-deleted subgraphs.
*Journal of Graph Theory* 67
(4),
pp. 285-299.
ISSN 0364-9024.

## Abstract

Let G be a graph of order n. The vertex-deleted subgraph G − v, obtained from G by deleting the vertex v and all edges incident to v, is called a card of G. Let H be another graph of order n, disjoint from G. Then the number of common cards of G and H is the maximum number of disjoint pairs (v, w), where v and w are vertices of G and H, respectively, such that G − v≅H − w. We prove that if G is connected and H is disconnected, then the number of common cards of G and H is at most ⌊n/2⌋ + 1. Thus, we can recognize the connectedness of a graph from any ⌊n/2⌋ + 2 of its cards. Moreover, we completely characterize those pairs of graphs that attain the upper bound and show that, with the exception of six pairs of graphs of order at most 7, any pair of graphs that attains the maximum is in one of four infinite families.

## Metadata

Item Type: | Article |
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Keyword(s) / Subject(s): | reconstruction conjecture, reconstruction number, card, vertex-deleted subgraph |

School: | Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences Birkbeck Faculties and Schools > Faculty of Business and Law > Birkbeck Business School |

Depositing User: | Administrator |

Date Deposited: | 17 Aug 2011 11:21 |

Last Modified: | 09 Aug 2023 12:30 |

URI: | https://eprints.bbk.ac.uk/id/eprint/4053 |

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